Multiplet structure of Feshbach resonances in nonzero partial waves

We report a unique feature of magnetic-field Feshbach resonances in which atoms collide with nonzero orbital angular momentum. $p$-wave $\left(l=1\right)$ Feshbach resonances are split into two components depending on the magnitude of the resonant state’s projection of orbital angular momentum onto the field axis. This splitting is due to the magnetic dipole-dipole interaction between the atoms and it offers a means to tune anisotropic interactions of an ultracold gas of atoms. Furthermore this splitting in the $p$-wave Feshbach resonance has been experimentally observed and is reported. A parametrization of the $p$-wave resonance in terms of an effective-range expansion is given.

Figure 1

(a) $p$-wave elastic cross section vs energy for $\mid 9∕2,-7∕2⟩\mid 9∕2,-7∕2⟩\mid 1,1⟩$ collisions for different magnetic-field values. For each curve the magnetic field in Gauss is indicated. The lowest curve shows an off-resonance cross section. (b) For comparison, the $s$-wave elastic cross section vs energy for $\mid 9∕2,-9∕2⟩\mid 9∕2,-7∕2⟩\mid 0,0⟩$ collisions for different magnetic-field values. The $s$-wave FR peaks at $B=201.6\mathrm{G}$. Compared to the $p$-wave FR these have little structure. The solid line is the unitarity limit.

Figure 2

(a) Thermally averaged cross section for $\mid 9∕2,-7∕2⟩\mid 9∕2,-7∕2⟩\mid 1,1⟩$ collisions as a function of magnetic field. The striking features of this curve are the sudden rise and change in width as the temperature is increased. (b) Thermally averaged cross section for $\mid 9∕2,-9∕2⟩\mid 9∕2,-7∕2⟩\mid 0,0⟩$ collisions as a function of magnetic field. The temperature dependence is only evident at the peak where it washes out the maximum value.

Figure 3

Schematic representation of classical dipoles interacting in different circular orbits. Shown in (a) is an orbit with $m_l=0$, which is in a plane containing the magnetic field. Here the dipoles sometimes attract and sometimes repel. In (b) is shown an orbit with $\mid m_l\mid =1$, in a plane perpendicular to the magnetic field. Here the atoms predominately repel one another.

Figure 4

The thermally averaged elastic cross section for the $p$-wave FR, including all partial-wave projections $m_l=-1,0,1$. At low temperatures, the doublet splitting emerges clearly, but it is washed out a higher temperatures due to thermal broadening. The lower field resonance has $\mid m_l\mid =1$ and the higher field resonance has $m_l=0$.

Figure 5

The $p$-wave FR observed through heating of the gas, clearly showing the doublet feature of the $p$-wave resonance. The cloud started at $T=0.34\mu \mathrm{K}$ and then was held at a constant magnetic field. Inelastic processes at the FR, three-body dominated, heat the cloud resulting in an increase in the measured size of the trapped cloud. The curve is only a guide to the eye.

Figure 6

(a) The $p$-wave scattering volume for $\mid m_l\mid =1$ as a function of magnetic field at two different energies. Notice that both the location at which the scattering volume diverges and the width vary with collision energy. (b) $k^3\cot \left({\delta }_1\right)$ for the $\mid m_l\mid =1$ $p$-wave resonance as a function of energy for several different values of magnetic field.